The center-mode instability of viscoelastic plane Poiseuille flow

Abstract

A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a `center mode' with phase speed close to the maximum base-flow velocity, $U_{max}$. The governing dimensionless groups are the Reynolds number $Re=\rho U_{max}H/\eta$, the elasticity number $E=\lambda \eta /(H^2\rho )$, and the ratio of solvent to solution viscosity ${\eta }_s/\eta$; here, $\lambda$ is the polymer relaxation time, $H$ is the channel half-width, and $\rho$ is the fluid density. For experimentally relevant values (e.g., $E\sim 0.1$ and $\beta \sim 0.9$), the predicted critical Reynolds number, $R{e}_c$, for the center-mode instability is around $200$, with the associated eigenmodes being spread out across the channel. In the asymptotic limit of $E(1-\beta )\ll 1$, with $E$ fixed, corresponding to strongly elastic dilute polymer solutions, $R{e}_c\propto (E(1-\beta ){)}^{-\frac{3}{2}}$ and the critical wavenumber $k_c\propto (E(1-\beta ){)}^{-\frac{1}{2}}$. The unstable eigenmode in this limit is confined in a thin layer near the channel centerline. The above features are largely analogous to the center-mode instability in viscoelastic pipe flow (Garg et al., Phys. Rev. Lett., 121, 024502 (2018)), and suggest a universal linear mechanism underlying the onset of turbulence in both channel and pipe flows of suffciently elastic dilute polymer solutions.

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• Understanding viscoelastic flow instabilities: Oldroyd-B and beyond
• The center-mode instability of viscoelastic plane Poiseuille flow
• A continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow
• Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers
• Tollmien-Schlichting route to elastoinertial turbulence in channel flow
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